Yes on a real basic level its cooling to prevent detonation. This is how I understand the whole concept. On a boosted setup you have two things to consider. How good is the engine at making power (moving air) and how good is the compressor is at compressing air (making power). Friction and drag will affect the motors power output at the shaft, and the efficiency of the compressor/intercooler will effect how much additional power you can add. The definition of brake specific fuel consumption is how efficient the setup is at using fuel to create a power output at the shaft.
"Lean is mean" if you can lean the afr safely you can make more power to a degree (this is creating more heat/energy during combustion to make that power). Since a bsfc rating is how efficient the setup is at using fuel to make power, then in turn you would be able to achieve a leaner afr or more efficient stoich ratio for that better bsfc setup. You can safely tune for it to make more power, they really go hand in hand. If you couldn't do this and every boosted setup had the exact same acheivable stoich to make the best power then they would all consume the same amount of fuel with the air to make that power.
So that's the other piece you mentioned, not all compressor have the same efficiency. Compressor efficiency is measured in how much heat it puts into the air while compressing the air. The hotter the air charge the less air there is per volume and the more fuel is needed to prevent detonation since that heat translates to more heat during combustion and fuel is what is used to cool things to prevent said detonation. The cooler the air is the denser it is per volume and less fuel is needed to keep things happy. So cooler air and leaner fuel makes more power. Notice I talked about air density and volume, this plays a role as well. This is why setup A might make 475rwhp at 10psi and setup B might make 525rwhp at 10psi. Your effectively moving more air at the same pressure. We like to measure compressor efficiency by this a lot. How much hp per psi does it produce. On a 300hp engine for both setups setup A is making 17.5 hp per psi and setup B is making 22.5 hp per psi. A belt driven compressor is always spinning and creating heat and you have to bypass this air when not wanted depending on engine load. Not the same for a load based compressor, when cruising low load there is no compressing taking place and no bypassing air so no heat being produced. We already talked about drag.
Last part is intercooler's efficiency is how well it can cool back down the heated air. A2w is more efficient than a2a. Pd/screw blowers always use the more efficient cooler as its what fits and goes better with the heat production. Centri/turbo setups most often have a2a as they can get away with the less efficient charge cooler as they produce less heat to compress the air.
My twin turbo setup was able to produce 610rwhp at 10psi on e85 which comes out to almost 30 hp per psi. How is this possible? Its all in the efficiency of the entire setup. Engine, fuel, compressor, intercooler. I have good ve in the motor with intake/cams and compression/light weight rotating assembly. The turbo setup and intercooler had ambient iat's so no additional heat was in the air. The increased volume of fuel used with e85 allows for additional cooling for leaner afr and more timing as well as the power produced from the more oxygen in said fuel.
Same way Travis is making a lot of power for his boost level.